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  <p>** Single 2D plane-strain quadrilateral element, subjected to sinusoidal base shaking**</p>
<ol class="arabic">
<li><p><strong>Tcl Code</strong></p>
<div class="highlight-tcl notranslate"><div class="highlight"><pre><span></span>#Created by Zhaohui Yang (zhyang@ucsd.edu)
#elastic pressure independent material
#plane strain,  single element,  dynamic analysis (input motion: sinusoidal acceleration at base)  
#SI units (m, s, KN, ton)
#
#         4     3
#         ------- 
#         |     |
#         |     |
#         |     |
#        1-------2   (nodes 1 and 2 fixed)
#         ^     ^
#          &lt;--&gt; input motion: sinusoidal acceleration at base
wipe
#
#some user defined variables
# 
set accMul   9.81     ;
set massDen  2.000      ;# solid mass density
set fluidDen 1.0        ;# fluid mass density
set massProportionalDamping   0.0 ;
set stiffnessProportionalDamping 0.001 ;
set cohesion 30 ;
set peakShearStrain 0.1 ;
set E1      90000.0      ;#Young&#39;s modulus
set poisson1 0.40 ;
set G [expr $E1/(2*(1+$poisson1))] ;
set B [expr $E1/(3*(1-2*$poisson1))] ;
set press    0        ;# isotropic consolidation pressure on quad element(s)
set period   1        ;# Period of applied sinusoidal load
set deltaT   0.01     ;# time step for analysis
set numSteps 2000     ;# Number of analysis steps
set gamma    0.5      ;# Newmark integration parameter
set pi 3.1415926535     ;
set inclination 0       ;
set unitWeightX [expr  ($massDen-$fluidDen)*9.81*sin($inclination/180.0*$pi)] ;# buoyant unit weight in X direction
set unitWeightY [expr -($massDen-$fluidDen)*9.81*cos($inclination/180.0*$pi)] ;# buoyant unit weight in Y direction
#############################################################

#create the ModelBuilder
model basic -ndm 2 -ndf 2

# define material and properties
nDMaterial PressureIndependMultiYield 2 2 $massDen $G $B  $cohesion $peakShearStrain
nDMaterial FluidSolidPorous 1 2 2 2.2e6

    
# define the nodes
node 1   0.0 0.0 
node 2   1.0 0.0 
node 3   1.0 1.0 
node 4   0.0 1.0

# define the element      thick  material      maTag   press  density  gravity 
element quad  1  1 2 3 4  1.0   &quot;PlaneStrain&quot;     2   $press  0.0    $unitWeightX  $unitWeightY  

updateMaterialStage -material 2 -stage 0

# fix the base in vertical direction
fix 1 1 1 
fix 2 1 1

#############################################################
# GRAVITY APPLICATION (elastic behavior)
# create the SOE, ConstraintHandler, Integrator, Algorithm and Numberer
system ProfileSPD
test NormDispIncr 1.e-12 25 0
constraints Transformation
integrator LoadControl 1 1 1 1
algorithm Newton 
numberer RCM

# create the Analysis
analysis Static

#analyze 
analyze 2

#############################################################
# NOW APPLY LOADING SEQUENCE AND ANALYZE (plastic)
# rezero time
setTime 0.0
wipeAnalysis

equalDOF 3 4   1 2    ;#tie nodes 3 and 4

# create a LoadPattern
pattern UniformExcitation 1 1 -accel &quot;Sine 0 10 $period -factor $accMul&quot;

# create the Analysis
constraints Penalty 1.0e18 1.0e18  ;# Transformation;  # 
test NormDispIncr 1.e-12 25 0
algorithm Newton 
numberer RCM
system ProfileSPD
rayleigh $massProportionalDamping 0.0 $stiffnessProportionalDamping 0.
integrator Newmark $gamma  [expr pow($gamma+0.5, 2)/4]  
analysis VariableTransient 

#create the recorder
recorder Node -file disp.out   -time  -node 1 2 3 4 -dof 1 2 -dT  0.01 disp
recorder Node -file acce.out  -time  -node 1 2 3 4 -dof 1 2 -dT 0.01 accel
recorder Element -ele 1 -time -file stress1.out -dT 0.01 material 1 stress 
recorder Element -ele 1 -time -file strain1.out -dT 0.01 material 1 strain 
recorder Element -ele 1 -time -file stress3.out -dT 0.01 material 3 stress 
recorder Element -ele 1 -time -file strain3.out -dT 0.01 material 3 strain 

#analyze 
set startT [clock seconds]
analyze $numSteps $deltaT [expr $deltaT/100] $deltaT 10
set endT [clock seconds]
puts &quot;Execution time: [expr $endT-$startT] seconds.&quot;

wipe  #flush ouput stream
</pre></div>
</div>
</li>
<li><p><strong>Python Code</strong></p></li>
<li><p><strong>Matlab Code</strong></p>
<div class="highlight-matlab notranslate"><div class="highlight"><pre><span></span><span class="n">clear </span><span class="s">all</span><span class="p">;</span>

<span class="n">a1</span><span class="p">=</span><span class="n">load</span><span class="p">(</span><span class="s">&#39;acce.out&#39;</span><span class="p">);</span>
<span class="n">d1</span><span class="p">=</span><span class="n">load</span><span class="p">(</span><span class="s">&#39;disp.out&#39;</span><span class="p">);</span>
<span class="n">s1</span><span class="p">=</span><span class="n">load</span><span class="p">(</span><span class="s">&#39;stress1.out&#39;</span><span class="p">);</span>
<span class="n">e1</span><span class="p">=</span><span class="n">load</span><span class="p">(</span><span class="s">&#39;strain1.out&#39;</span><span class="p">);</span>
<span class="n">s5</span><span class="p">=</span><span class="n">load</span><span class="p">(</span><span class="s">&#39;stress3.out&#39;</span><span class="p">);</span>
<span class="n">e5</span><span class="p">=</span><span class="n">load</span><span class="p">(</span><span class="s">&#39;strain3.out&#39;</span><span class="p">);</span>

<span class="n">fs</span><span class="p">=[</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">6</span><span class="p">];</span>
<span class="n">accMul</span> <span class="p">=</span> <span class="mf">9.81</span><span class="p">;</span>

<span class="c">%integration point 1 p-q</span>
<span class="n">po</span><span class="p">=(</span><span class="n">s1</span><span class="p">(:,</span><span class="mi">2</span><span class="p">)</span><span class="o">+</span><span class="n">s1</span><span class="p">(:,</span><span class="mi">3</span><span class="p">)</span><span class="o">+</span><span class="n">s1</span><span class="p">(:,</span><span class="mi">4</span><span class="p">))</span><span class="o">/</span><span class="mi">3</span><span class="p">;</span><span class="n"></span>
<span class="n">for </span><span class="s">i=1:size(s1,1)</span><span class="p"></span>
	<span class="n">qo</span><span class="p">(</span><span class="nb">i</span><span class="p">)=(</span><span class="n">s1</span><span class="p">(</span><span class="nb">i</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span><span class="o">-</span><span class="n">s1</span><span class="p">(</span><span class="nb">i</span><span class="p">,</span><span class="mi">3</span><span class="p">))</span>^<span class="mi">2</span> <span class="o">+</span> <span class="p">(</span><span class="n">s1</span><span class="p">(</span><span class="nb">i</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span><span class="o">-</span><span class="n">s1</span><span class="p">(</span><span class="nb">i</span><span class="p">,</span><span class="mi">4</span><span class="p">))</span>^<span class="mi">2</span> <span class="o">+</span><span class="p">(</span><span class="n">s1</span><span class="p">(</span><span class="nb">i</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span><span class="o">-</span><span class="n">s1</span><span class="p">(</span><span class="nb">i</span><span class="p">,</span><span class="mi">4</span><span class="p">))</span>^<span class="mi">2</span> <span class="o">+</span> <span class="mf">6.0</span><span class="o">*</span> <span class="n">s1</span><span class="p">(</span><span class="nb">i</span><span class="p">,</span><span class="mi">5</span><span class="p">)</span>^<span class="mi">2</span><span class="p">;</span>
<span class="n">qo</span><span class="p">(</span><span class="nb">i</span><span class="p">)=</span><span class="nb">sign</span><span class="p">(</span><span class="n">s1</span><span class="p">(</span><span class="nb">i</span><span class="p">,</span><span class="mi">5</span><span class="p">))</span><span class="o">*</span><span class="mi">1</span><span class="o">/</span><span class="mf">3.0</span><span class="o">*</span><span class="n">qo</span><span class="p">(</span><span class="nb">i</span><span class="p">)</span>^<span class="mf">0.5</span><span class="p">;</span><span class="n"></span>
<span class="n">end</span>
<span class="s">figure(1)</span><span class="p">;</span> <span class="n">clf</span><span class="p">;</span>
<span class="c">%integration point 1 stress-strain</span>
<span class="n">subplot</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">),</span> <span class="n">plot</span><span class="p">(</span><span class="n">e1</span><span class="p">(:,</span><span class="mi">4</span><span class="p">),</span><span class="n">s1</span><span class="p">(:,</span><span class="mi">5</span><span class="p">),</span><span class="s">&#39;r&#39;</span><span class="p">);</span>
<span class="n">title</span> <span class="p">(</span><span class="s">&#39;Integration point 1 shear stress \tau_x_y VS. shear strain \epsilon_x_y&#39;</span><span class="p">);</span>
<span class="n">xLabel</span><span class="p">(</span><span class="s">&#39;Shear strain \epsilon_x_y&#39;</span><span class="p">);</span>
<span class="n">yLabel</span><span class="p">(</span><span class="s">&#39;Shear stress \tau_x_y (kPa)&#39;</span><span class="p">);</span>

<span class="n">subplot</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">),</span> <span class="n">plot</span><span class="p">(</span><span class="o">-</span><span class="n">po</span><span class="p">,</span><span class="n">qo</span><span class="p">,</span><span class="s">&#39;r&#39;</span><span class="p">);</span>
<span class="n">title</span> <span class="p">(</span><span class="s">&#39;Integration point 1 confinement p VS. deviatoric q relation&#39;</span><span class="p">);</span>
<span class="n">xLabel</span><span class="p">(</span><span class="s">&#39;confinement p (kPa)&#39;</span><span class="p">);</span>
<span class="n">yLabel</span><span class="p">(</span><span class="s">&#39;q (kPa)&#39;</span><span class="p">);</span>
<span class="n">set</span><span class="p">(</span><span class="n">gcf</span><span class="p">,</span><span class="s">&#39;paperposition&#39;</span><span class="p">,</span><span class="n">fs</span><span class="p">);</span>
<span class="n">saveas</span><span class="p">(</span><span class="n">gcf</span><span class="p">,</span><span class="s">&#39;SS_PQ1&#39;</span><span class="p">,</span><span class="s">&#39;jpg&#39;</span><span class="p">);</span>

<span class="c">%integration point 3 p-q</span>
<span class="n">po</span><span class="p">=(</span><span class="n">s5</span><span class="p">(:,</span><span class="mi">2</span><span class="p">)</span><span class="o">+</span><span class="n">s5</span><span class="p">(:,</span><span class="mi">3</span><span class="p">)</span><span class="o">+</span><span class="n">s5</span><span class="p">(:,</span><span class="mi">4</span><span class="p">))</span><span class="o">/</span><span class="mi">3</span><span class="p">;</span><span class="n"></span>
<span class="n">for </span><span class="s">i=1:size(s5,1)</span><span class="p"></span>
	<span class="n">qo</span><span class="p">(</span><span class="nb">i</span><span class="p">)=(</span><span class="n">s5</span><span class="p">(</span><span class="nb">i</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span><span class="o">-</span><span class="n">s5</span><span class="p">(</span><span class="nb">i</span><span class="p">,</span><span class="mi">3</span><span class="p">))</span>^<span class="mi">2</span> <span class="o">+</span> <span class="p">(</span><span class="n">s5</span><span class="p">(</span><span class="nb">i</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span><span class="o">-</span><span class="n">s5</span><span class="p">(</span><span class="nb">i</span><span class="p">,</span><span class="mi">4</span><span class="p">))</span>^<span class="mi">2</span> <span class="o">+</span><span class="p">(</span><span class="n">s5</span><span class="p">(</span><span class="nb">i</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span><span class="o">-</span><span class="n">s5</span><span class="p">(</span><span class="nb">i</span><span class="p">,</span><span class="mi">4</span><span class="p">))</span>^<span class="mi">2</span> <span class="o">+</span> <span class="mf">6.0</span><span class="o">*</span> <span class="n">s5</span><span class="p">(</span><span class="nb">i</span><span class="p">,</span><span class="mi">5</span><span class="p">)</span>^<span class="mi">2</span><span class="p">;</span>
<span class="n">qo</span><span class="p">(</span><span class="nb">i</span><span class="p">)=</span><span class="nb">sign</span><span class="p">(</span><span class="n">s5</span><span class="p">(</span><span class="nb">i</span><span class="p">,</span><span class="mi">5</span><span class="p">))</span><span class="o">*</span><span class="mi">1</span><span class="o">/</span><span class="mf">3.0</span><span class="o">*</span><span class="n">qo</span><span class="p">(</span><span class="nb">i</span><span class="p">)</span>^<span class="mf">0.5</span><span class="p">;</span><span class="n"></span>
<span class="n">end</span>

<span class="s">figure(4)</span><span class="p">;</span> <span class="n">clf</span><span class="p">;</span>
<span class="c">%integration point 3 stress-strain</span>
<span class="n">subplot</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">),</span> <span class="n">plot</span><span class="p">(</span><span class="n">e5</span><span class="p">(:,</span><span class="mi">4</span><span class="p">),</span><span class="n">s5</span><span class="p">(:,</span><span class="mi">5</span><span class="p">),</span><span class="s">&#39;r&#39;</span><span class="p">);</span>
<span class="n">title</span> <span class="p">(</span><span class="s">&#39;Integration point 3 shear stress \tau_x_y VS. shear strain \epsilon_x_y&#39;</span><span class="p">);</span>
<span class="n">xLabel</span><span class="p">(</span><span class="s">&#39;Shear strain \epsilon_x_y&#39;</span><span class="p">);</span>
<span class="n">yLabel</span><span class="p">(</span><span class="s">&#39;Shear stress \tau_x_y (kPa)&#39;</span><span class="p">);</span>

<span class="n">subplot</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">),</span> <span class="n">plot</span><span class="p">(</span><span class="o">-</span><span class="n">po</span><span class="p">,</span><span class="n">qo</span><span class="p">,</span><span class="s">&#39;r&#39;</span><span class="p">);</span>
<span class="n">title</span> <span class="p">(</span><span class="s">&#39;Integration point 3 confinement p VS. deviatoric q relation&#39;</span><span class="p">);</span>
<span class="n">xLabel</span><span class="p">(</span><span class="s">&#39;confinement p (kPa)&#39;</span><span class="p">);</span>
<span class="n">yLabel</span><span class="p">(</span><span class="s">&#39;q (kPa)&#39;</span><span class="p">);</span>
<span class="n">set</span><span class="p">(</span><span class="n">gcf</span><span class="p">,</span><span class="s">&#39;paperposition&#39;</span><span class="p">,</span><span class="n">fs</span><span class="p">);</span>
<span class="n">saveas</span><span class="p">(</span><span class="n">gcf</span><span class="p">,</span><span class="s">&#39;SS_PQ5&#39;</span><span class="p">,</span><span class="s">&#39;jpg&#39;</span><span class="p">);</span>

<span class="n">figure</span><span class="p">(</span><span class="mi">2</span><span class="p">);</span> <span class="n">clf</span><span class="p">;</span>
<span class="c">%node 3 displacement relative to node 1</span>
<span class="n">subplot</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">),</span><span class="n">plot</span><span class="p">(</span><span class="n">d1</span><span class="p">(:,</span><span class="mi">1</span><span class="p">),</span><span class="n">d1</span><span class="p">(:,</span><span class="mi">6</span><span class="p">),</span><span class="s">&#39;r&#39;</span><span class="p">);</span>
<span class="n">title</span> <span class="p">(</span><span class="s">&#39;Lateral displacement at element top&#39;</span><span class="p">);</span>
<span class="n">xLabel</span><span class="p">(</span><span class="s">&#39;Time (s)&#39;</span><span class="p">);</span>
<span class="n">yLabel</span><span class="p">(</span><span class="s">&#39;Displacement (m)&#39;</span><span class="p">);</span> 
<span class="n">set</span><span class="p">(</span><span class="n">gcf</span><span class="p">,</span><span class="s">&#39;paperposition&#39;</span><span class="p">,</span><span class="n">fs</span><span class="p">);</span>
<span class="n">saveas</span><span class="p">(</span><span class="n">gcf</span><span class="p">,</span><span class="s">&#39;D&#39;</span><span class="p">,</span><span class="s">&#39;jpg&#39;</span><span class="p">);</span>


<span class="n">s</span><span class="p">=</span><span class="n">accMul</span><span class="o">*</span><span class="nb">sin</span><span class="p">(</span><span class="mi">0</span><span class="p">:</span><span class="nb">pi</span><span class="o">/</span><span class="mi">50</span><span class="p">:</span><span class="mi">20</span><span class="o">*</span><span class="nb">pi</span><span class="p">);</span>
<span class="n">s</span><span class="p">=[</span><span class="n">s</span><span class="o">&#39;</span><span class="p">;</span><span class="nb">zeros</span><span class="p">(</span><span class="mi">1000</span><span class="p">,</span><span class="mi">1</span><span class="p">)];</span>
<span class="n">s1</span><span class="p">=</span><span class="n">interp1</span><span class="p">(</span><span class="mi">0</span><span class="p">:</span><span class="mf">0.01</span><span class="p">:</span><span class="mi">20</span><span class="p">,</span><span class="n">s</span><span class="p">,</span><span class="n">a1</span><span class="p">(:,</span><span class="mi">1</span><span class="p">));</span>

<span class="n">figure</span><span class="p">(</span><span class="mi">3</span><span class="p">);</span> <span class="n">clf</span><span class="p">;</span>
<span class="c">%node 3 relative acceleration</span>
<span class="n">subplot</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">),</span><span class="n">plot</span><span class="p">(</span><span class="n">a1</span><span class="p">(:,</span><span class="mi">1</span><span class="p">),</span><span class="n">s1</span><span class="o">+</span><span class="n">a1</span><span class="p">(:,</span><span class="mi">5</span><span class="p">),</span><span class="s">&#39;r&#39;</span><span class="p">);</span>
<span class="n">title</span> <span class="p">(</span><span class="s">&#39;Lateral acceleration at element top&#39;</span><span class="p">);</span>
<span class="n">xLabel</span><span class="p">(</span><span class="s">&#39;Time (s)&#39;</span><span class="p">);</span>
<span class="n">yLabel</span><span class="p">(</span><span class="s">&#39;Acceleration (m/s^2)&#39;</span><span class="p">);</span>
<span class="n">set</span><span class="p">(</span><span class="n">gcf</span><span class="p">,</span><span class="s">&#39;paperposition&#39;</span><span class="p">,</span><span class="n">fs</span><span class="p">);</span>
<span class="n">saveas</span><span class="p">(</span><span class="n">gcf</span><span class="p">,</span><span class="s">&#39;A&#39;</span><span class="p">,</span><span class="s">&#39;jpg&#39;</span><span class="p">);</span>
</pre></div>
</div>
</li>
</ol>


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